GUJARAT TECHNOLOGICAL UNIVERSITY
SEMESTER- IV (New) EXAMINATION — WINTER 2019
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Subject Code: 2141905 Date: 07/12/2019
Subject Name: Complex Variables and Numerical Methods
Time: 10:30 AM TO 01:30 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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Q-1
(a) Determine whether the function is continuous? Can the function be redefined to make it continuous at Z = —i? [03]
(b) State De Moivre’s Theorem. Find the roots of the equation Z4+1=0. [04]
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(c) Solve the following system of equations using Gauss Seidel Method correct to four decimal places. [07]
30x —2y+3z=75; 2x+2y+18z=30; x + 17y — 2z = 48
Q-2
(a) Check whether the function f (z) = ex is entire or not. Also find derivative of f(z). [03]
(b) Find the bilinear transformation which maps z = 1, 0, —1 into the points W = i, 1. [04]
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(c) Using the Residue Theorem Evaluate, ? 0 2p d? / (5 + 3sin?) [07]
OR
(c) Show that the function u(x,y) = 3x2y + 2x2 — y3 — 2y2 is harmonic. Find the conjugate harmonic function v and express u + iv as analytic function of z [07]
Q-3
(a) Evaluate ?c (z2+1)/ (z-1) dZ if c is the circle of unit radius with centre at z = 1. [03]
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(b) Find the real part and imaginary part of (1/z). [04]
(c) Evaluate ? f(z)dz where f(z) is defined by [07]
f(z) = { 1 when y < 0; 4 when y > 0 }
And C is the arc from z = —1 —i to z =1+ i along the curve y = x3.
OR
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(c) Find the type of singularity of the function f(z) = e2z / (z2(z-1)4) [07]
Q-4
(a) Expand f(Z) = 1/((Z-1)(Z-2)) valid for region [03]
(i) |z| < 1 (ii) 1 < |z| < 2 (iii) |z| > 2
(b) Use Euler’s Method, find y(0.2) given that dy/dx = x - y2 ; y(0)=1 take h = 0.1 [04]
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(c) Evaluate v8 to two decimal places by Newton’s iterative formula. [07]
OR
(a) Solve the following system of equation using Gauss Elimination Method [03]
x+y+z=7, 3x+3y+4z=24; 2x+y+3z=16
(b) Use Secant Method to find the root of f(x) = xlog10x —1.9 =0 [04]
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(c) Using Newton’s Divided Differences formula to find a polynomial function, satisfying the following data. [07]
| x | -4 | -1 | 0 | 2 | 5 |
|---|---|---|---|---|---|
| f(x) | 1245 | 33 | 5 | 9 | 1335 |
Q-5
(a) Evaluate ?12 1/x dx by using Gaussian formula for n =2 and n = 3 [03]
(b) Use fourth order Range-Kutta method to compute y(0.2) and y(0.4) given that dy/dx = y - x2 ; y(0) = 1 [04]
(c) Find the dominant Eigen value of A = [[4, -5], [-3, 4]] by Power Method and the corresponding Eigen vector. [07]
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OR
(a) State Trapezoidal Rule and evaluate ?01 e-x2 dx using it with n = 10 [03]
(b) Use Lagrange’s formula to fit a polynomial to the data [04]
| X | -1 | 0 | 2 | 3 |
|---|---|---|---|---|
| Y | 8 | 3 | 1 | 12 |
(c) Apply improved Euler’s method to solve the initial value problem y' = x +y with y(0) = 0 choosing h = 0.2 and compute y1, y2, ..., y5. [07]
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